Generalized Nonlinear Yule Models

TL;DR

This paper introduces a fractional nonlinear Yule model to better capture persistent memory effects in network growth, providing explicit distributions and mean values for in-link counts, with applications to web development and natural phenomena.

Contribution

It develops a generalized fractional nonlinear Yule model with explicit distributions and mean calculations, extending classical models to include nonlinear and memory effects.

Findings

Derived explicit distribution of in-links for the model

Calculated mean in-link count in the general case

Showed saturating behavior in specific birth rate cases

Abstract

With the aim of considering models with persistent memory we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macrovolution. Here the model is analyzed and interpreted in the framework of the development of networks such as the World Wide Web. Nonlinearity is introduced by replacing the linear birth process governing the growth of the in-links of each specific webpage with a fractional nonlinear birth process with completely general birth rates. Among the main results we derive the explicit distribution of the number of in-links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction. The mean value of the latter distribution is also calculated explicitly in the most general case. Furthermore, in order to show the usefulness of our results, we particularize them in the case of specific birth rates giving rise to a saturating behaviour, a property that is often observed in nature. The further specialization to the non-fractional case allows us to extend the Yule model accounting for a nonlinear growth.